"...The authors of this remarkable book are among the very few whohave faced up to the challenge of explaining what an asymptoticexpansion is, and of systematizing the handling of asymptotic series.The idea of using distributions is an original one, and we recommendthat you read the book...[it] should be on your bookshelf if you areat all interested in knowing what an asymptotic series is." -"TheBulletin of Mathematics Books" (Review of the 1st edition) __"...The book is a valuable one, one that many applied mathematiciansmay want to buy. The authors are undeniably experts in theirfield...most…mehr
"...The authors of this remarkable book are among the very few whohave faced up to the challenge of explaining what an asymptoticexpansion is, and of systematizing the handling of asymptotic series.The idea of using distributions is an original one, and we recommendthat you read the book...[it] should be on your bookshelf if you areat all interested in knowing what an asymptotic series is." -"TheBulletin of Mathematics Books" (Review of the 1st edition) __"...The book is a valuable one, one that many applied mathematiciansmay want to buy. The authors are undeniably experts in theirfield...most of the material has appeared in no other book." -"SIAMNews" (Review of the 1st edition)This book is a modern introduction to asymptotic analysis intendednot only for mathematicians, but for physicists, engineers, andgraduate students as well. Written by two of the leading experts inthe field, the text provides readers with a firm grasp of mathematicaltheory, and at the same time demonstrates applications in areas suchas differential equations, quantum mechanics, noncommutative geometry,and number theory.Key features of this significantly expanded and revised secondedition: _ addition of a new chapter and many new sections _ widerange of topics covered, including the Ces.ro behavior ofdistributions and their connections to asymptotic analysis, the studyof time-domain asymptotics, and the use of series of Dirac deltafunctions to solve boundary value problems _ novel approach detailingthe interplay between underlying theories of asymptotic analysis andgeneralized functions _ extensive examples and exercises at the end ofeach chapter _ comprehensive bibliography and indexThis work is an excellent tool for the classroom and an invaluableself-study resource that will stimulate application of asymptotic
Estrada, R., Universidad de Costa Rica, San Jose, Costa Rica / Kanwal, R.P., Pennsylvania State University, USA
Inhaltsangabe
1 Basic Results in Asymptotics.- 1.1 Introduction.- 1.2 Order Symbols.- 1.3 Asymptotic Series.- 1.4 Algebraic and Analytic Operations.- 1.5 Existence of Functions with a Given Asymptotic Expansion.- 1.6 Asymptotic Power Series in a Complex Variable.- 1.7 Asymptotic Approximation of Partial Sums.- 1.8 The Euler-Maclaurin Summation Formula.- 1.9 Exercises.- 2 Introduction to the Theory of Distributions.- 2.1 Introduction.- 2.2 The Space of Distributions D?.- 2.3 Algebraic and Analytic Operations.- 2.4 Regularization, Pseudofunction and Hadamard Finite Part.- 2.5 Support and Order.- 2.6 Homogeneous Distributions.- 2.7 Distributional Derivatives of Discontinuous Functions.- 2.8 Tempered Distributions and the Fourier Transform.- 2.9 Distributions of Rapid Decay.- 2.10 Spaces of Distributions Associated with an Asymptotic Sequence.- 2.11 Exercises.- 3 A Distributional Theory for Asymptotic Expansions.- 3.1 Introduction.- 3.2 The Taylor Expansion of Distributions.- 3.3 The Moment Asymptotic Expansion.- 3.4 Expansions in the Space P?.- 3.5 Laplace's Asymptotic Formula.- 3.6 The Method of Steepest Descent.- 3.7 Expansion of Oscillatory Kernels.- 3.8 Time-Domain Asymptotics.- 3.9 The Expansion of f (?x) as ? ? ? in Other Cases.- 3.10 Asymptotic Separation of Variables.- 3.11 Exercises.- 4 Asymptotic Expansion of Multidimensional Generalized Functions.- 4.1 Introduction.- 4.2 Taylor Expansion in Several Variables.- 4.3 The Multidimensional Moment Asymptotic Expansion.- 4.4 Laplace's Asymptotic Formula.- 4.5 Fourier Type Integrals.- 4.6 Time-Domain Asymptotics.- 4.7 Further Examples.- 4.8 Tensor Products and Partial Asymptotic Expansions.- 4.9 An Application in Quantum Mechanics.- 4.10 Expansion of Kernels of the Type f (?x, x).- 4.11 Exercises.- 5 AsymptoticExpansion of Certain Series Considered by Ramanujan.- 5.1 Introduction.- 5.2 Basic Formulas.- 5.3 Lambert Type Series.- 5.4 Distributionally Small Sequences.- 5.5 Multiple Series.- 5.6 Unrestricted Partitions.- 5.7 Exercises.- 6 Cesàro Behavior of Distributions.- 6.1 Introduction.- 6.2 Summability of Series and Integrals.- 6.3 The Behavior of Distributions in the (C) Sense.- 6.4 The Cesàro Summability of Evaluations.- 6.5 Parametric Behavior.- 6.6 Characterization of Tempered Distributions.- 6.7 The Space K?.- 6.8 Spherical Means.- 6.9 Existence of Regularizations.- 6.10 The Integral Test.- 6.11 Moment Functions.- 6.12 The Analytic Continuation of Zeta Functions.- 6.13 Fourier Series.- 6.14 Summability of Trigonometric Series.- 6.15 Distributional Point Values of Fourier Series.- 6.16 Spectral Asymptotics.- 6.17 Pointwise and Average Expansions.- 6.18 Global Expansions.- 6.19 Asymptotics of the Coincidence Limit.- 6.20 Exercises.- 7 Series of Dirac Delta Functions.- 7.1 Introduction.- 7.2 Basic Notions.- 7.3 Several Problems that Lead to Series of Deltas.- 7.4 Dual Taylor Series as Asymptotics of Solutions of Equations.- 7.5 Boundary Layers.- 7.6 Spectral Content Asymptotics.- 7.7 Exercises.- References.
1 Basic Results in Asymptotics.- 1.1 Introduction.- 1.2 Order Symbols.- 1.3 Asymptotic Series.- 1.4 Algebraic and Analytic Operations.- 1.5 Existence of Functions with a Given Asymptotic Expansion.- 1.6 Asymptotic Power Series in a Complex Variable.- 1.7 Asymptotic Approximation of Partial Sums.- 1.8 The Euler-Maclaurin Summation Formula.- 1.9 Exercises.- 2 Introduction to the Theory of Distributions.- 2.1 Introduction.- 2.2 The Space of Distributions D?.- 2.3 Algebraic and Analytic Operations.- 2.4 Regularization, Pseudofunction and Hadamard Finite Part.- 2.5 Support and Order.- 2.6 Homogeneous Distributions.- 2.7 Distributional Derivatives of Discontinuous Functions.- 2.8 Tempered Distributions and the Fourier Transform.- 2.9 Distributions of Rapid Decay.- 2.10 Spaces of Distributions Associated with an Asymptotic Sequence.- 2.11 Exercises.- 3 A Distributional Theory for Asymptotic Expansions.- 3.1 Introduction.- 3.2 The Taylor Expansion of Distributions.- 3.3 The Moment Asymptotic Expansion.- 3.4 Expansions in the Space P?.- 3.5 Laplace's Asymptotic Formula.- 3.6 The Method of Steepest Descent.- 3.7 Expansion of Oscillatory Kernels.- 3.8 Time-Domain Asymptotics.- 3.9 The Expansion of f (?x) as ? ? ? in Other Cases.- 3.10 Asymptotic Separation of Variables.- 3.11 Exercises.- 4 Asymptotic Expansion of Multidimensional Generalized Functions.- 4.1 Introduction.- 4.2 Taylor Expansion in Several Variables.- 4.3 The Multidimensional Moment Asymptotic Expansion.- 4.4 Laplace's Asymptotic Formula.- 4.5 Fourier Type Integrals.- 4.6 Time-Domain Asymptotics.- 4.7 Further Examples.- 4.8 Tensor Products and Partial Asymptotic Expansions.- 4.9 An Application in Quantum Mechanics.- 4.10 Expansion of Kernels of the Type f (?x, x).- 4.11 Exercises.- 5 AsymptoticExpansion of Certain Series Considered by Ramanujan.- 5.1 Introduction.- 5.2 Basic Formulas.- 5.3 Lambert Type Series.- 5.4 Distributionally Small Sequences.- 5.5 Multiple Series.- 5.6 Unrestricted Partitions.- 5.7 Exercises.- 6 Cesàro Behavior of Distributions.- 6.1 Introduction.- 6.2 Summability of Series and Integrals.- 6.3 The Behavior of Distributions in the (C) Sense.- 6.4 The Cesàro Summability of Evaluations.- 6.5 Parametric Behavior.- 6.6 Characterization of Tempered Distributions.- 6.7 The Space K?.- 6.8 Spherical Means.- 6.9 Existence of Regularizations.- 6.10 The Integral Test.- 6.11 Moment Functions.- 6.12 The Analytic Continuation of Zeta Functions.- 6.13 Fourier Series.- 6.14 Summability of Trigonometric Series.- 6.15 Distributional Point Values of Fourier Series.- 6.16 Spectral Asymptotics.- 6.17 Pointwise and Average Expansions.- 6.18 Global Expansions.- 6.19 Asymptotics of the Coincidence Limit.- 6.20 Exercises.- 7 Series of Dirac Delta Functions.- 7.1 Introduction.- 7.2 Basic Notions.- 7.3 Several Problems that Lead to Series of Deltas.- 7.4 Dual Taylor Series as Asymptotics of Solutions of Equations.- 7.5 Boundary Layers.- 7.6 Spectral Content Asymptotics.- 7.7 Exercises.- References.
Rezensionen
"This is not just a 'Second Edition' of some monograph in the usual sense, but a revised and largely expanded version of Asymptotic Analysis: A Distributional Approach (1994) by the same authors.... A completely new chapter on the Cesáro behavior of distributions has been added; moreover there are several new sections, among them respective problem sections at the end of each chapter. Finally, a large number of recent results and additional examples have been included.... Even more than its predecessor, this book presents an interesting and carefully written introduction into the theory and applications of asymptotic analysis based on distribution theory." -MONATSHEFTE FÜR MATHEMATIK (Review of the Second Edition) "The authors of this remarkable book are among the very few that have faced up to the challenge of explaining what an asymptotic expansion is, and of systematizing the handling of asymptotic series. The idea of using distributions is an original one, and we recommend that you read the book...[it] should be on your bookshelf if you are at all interested in knowing what an asymptotic series is." -THE BULLETIN OF MATHEMATICS BOOKS (Review of the First Edition) ". . . the book is a valuable one, one that many applied mathematicians may want to buy. The authors are undeniably experts in their field . . . most of the material has appeared in no other book." -SIAM REVIEW (Review of the First Edition)
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